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Factorial
The factorial is a function applied to whole numbers, defined asFactorial from Wolfram MathWorldFactorials from PurpleMath $$n! = \prod^n_{i = 1} i = n \cdot (n - 1) \cdot \ldots \cdot 4 \cdot 3 \cdot 2 \cdot 1.$$ For example, \(6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720\). It is equal to the number of ways \(n\) distinct objects can be arranged, because there are \(n\) ways to place the first object, \(n - 1\) ways to place the second object, and so forth. The special case \(0! = 1\) has been set by definition; there is one way to arrange zero objects. Before the notation \(n!\) was invented, \(n\) was common. The function can be defined recursively as \(0! = 1\) and \(n! = n \cdot (n - 1)!\). The first few values of \(n!\) for \(n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\) are 1, 1 , 2, 6, 24, 120, 720, 5,040, 40,320, 362,880, 3,628,800, and 39,916,800. Properties The sum of the s of the factorials is \(\sum^{\infty}_{i = 0} \frac{1}{i!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots = 2.71828182845904\ldots\), a mathematical constant better known as \(e\). In fact, \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\), which illustrates the important property that \(\frac{d}{dx}e^x = e^x\). Because \(n! = \Gamma (n + 1)\) (where \(\Gamma (x)\) is the ), \(n! = \int^{\infty}_0 e^{-t} \cdot t^{n} dt\). This identity gives us factorials of positive real numbers (and negative non-integer real numbers), not limited to integers: *\(\left(\frac{1}{2}\right)! = \frac{\sqrt{\pi}}{2}\) *\(\left(-\frac{1}{2}\right)! = \sqrt{\pi}\) The most well-known approximation of n! is \(n!\approx \sqrt{2\pi n}(\frac{n}{e})^n\), and it's called . In base 10, only two non-trivial numbers are equal to the sum of the factorials of their digits: \(145 = 1! + 4! + 5! = 5 × 29\) and \(40,585 = 4! + 0! + 5! + 8! + 5! = 5 × 8,117\). The number of zeroes at the end of the decimal expansion of \(n!\) is \(\sum_{k = 1} \lfloor n / 5^k\rfloor\).Factorials and Trailing Zeroes from PurpleMath For example, 10,000! has 2,000 + 400 + 80 + 16 + 3 = 2,499 zeroes. Specific numbers *'153' is the sum of the factorials of the first five positive numbers, and also the exponent in the short scale quinquagintillion. **The first carrier frequency in the longwave radio band is at 153 kHz. **It is also the number of fish in the second . *'154' is the sum of the factorials of the first six nonnegative numbers, and 154! + 1 is a . **It is also a and the 7th . **Its prime factorization is 2 × 7 × 11.https://www.wolframalpha.com/input/?i=154 *'720' is an integer equal to 6!, the factorial of 6. Consequently, it is the order of the of degree 6, which is isomorphic to , and has an . **It is also the number of degrees in a hexagon. **Furthermore, it is the number of hours in a 30-day month (April, June, September or November) not containing a transition. **And some services have 720 visible scan lines. **Finally, it is also the number of pixels in a standard-definition television scan line. *'5,040' is an integer equal to 7!. It is the largest known factorial number which is the predecessor of a square number: 7! = 5,040 = 5,041−1 = 712−1. ** mentioned in his that 5,040 is a convenient number to use for dividing many things (including both the citizens and the land of a city-state or polis) into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a polis. He remarks that this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11 (however, it is not the smallest number to have this property; 2,520 is). *'479,001,600' is equal to \(12!\), and therefore the number of possible tone rows in the . *'1,124,000,727,777,607,680,000' is a positive integer equal to \(22!\). It is notable in computer science for being the largest factorial number which can be represented exactly in the double floating-point format (which has a 53-bit significand). **In the short scale, this number is written as 1 sextillion 124 quintillion 727 trillion 777 billion 607 million 680 thousand. **In the long scale, this number is written as 1 trilliard 124 trillion 727 billion 777 milliard 607 million 680 thousand. * 70! is the smallest factorial which is greater than googol, while 69! still has only 99 digits. * One hundred factorial's decimal expansion is shown below. *: **In scientific notation, this is approximately 9.3326215443 × . It seems to be approximately , although it is almost 100 million times larger. *Lawrence Hollom calls 200! faxul. *One thousand factorial is about 4.0238726007 × . *Aarex Tiaokhiao has proposed the name Myriadbang for 10,000!. *One million factorial is approximately 8.2639317 × 105,565,708. *One billion factorial is approximately 1.57637137 × 108,565,705,531. Variation Aalbert Torsius defines a variation on the factorial, where \(x!n = \prod^{x}_{i = 1} i!(n - 1) = 1!(n - 1) \cdot 2!(n - 1) \cdot \ldots \cdot x!(n - 1)\) and \(x!0 = x\).http://c2.com/cgi/wiki?ReallyBigNumbers \(x!n\) is pronounced "n''th level factorial of ''x." \(x!1\) is simply the ordinary factorial and \(x!2\) is Sloane and Plouffe's superfactorial \(x\$\). The special case \(x!x\) is a function known as the Torian. Pseudocode // Standard factorial function function factorial(z''): ''result := 1 for i'' '''from' 1 to z'': ''result := result * i'' '''return' result // Generalized factorial, using Lanczos approximation for gamma function g'' := 7 ''coeffs := 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 function factorialReal(z''): ''ag := coeffs0 for i'' '''from' 1 to g'' + 1: ''ag := ag + coeffs[i''] / (''z + i'') ''zg := z'' + ''g + 0.5 return sqrt(2 * pi) * * * ag // Torsius' factorial extension function factorialTorsius(z'', ''x): if x'' = 0: '''return' z'' '''if' x'' = 1: '''return' factorial(z'') ''result := 1 for i'' '''from' 1 to z'': ''result := result * factorialTorsius(i'', ''x - 1) return result Sources See also de:Fakultät ja:階乗 nl:Faculteit zh:階乘 Category:Functions Category:Factorials Category:Combinatorics Category:Notations